scholarly journals Complex dynamics in an SIS epidemic model induced by nonlinear incidence

2018 ◽  
Author(s):  
Ruixia Yuan ◽  
Zhidong Teng ◽  
Jinhui Li
Keyword(s):  
Epidemic Model ◽  
Complex Dynamics ◽  
Psychological Effect ◽  
Mass Action ◽  
Phase Portraits ◽  
Nonlinear Incidence Rate ◽  
Homoclinic Loop ◽  
Nonlinear Incidence ◽  
Sis Epidemic Model ◽  
Disease Free

AbstractWe study an epidemic model with nonlinear incidence rate, describing the saturated mass action as well as the psychological effect of certain serious diseases on the community. Firstly, the existence and local stability of disease-free and endemic equilibria are investigated. Then, we prove the occurrence of backward bifurcation, saddle-node bifurcation, Hopf bifurcation and cusp type Bogdanov-Takens bifurcation of codimension 3. Finally, numerical simulations, including one limit cycle, two limit cycles, unstable homoclinic loop and many other phase portraits are presented. These results show that the psychological effect of diseases and the behavior change of the susceptible individuals may affect the final spread level of an epidemic.

ANALYSIS ON AN EPIDEMIC MODEL WITH A RATIO-DEPENDENT NONLINEAR INCIDENCE RATE

2011 ◽  
Vol 04 (02) ◽  
pp. 227-239 ◽  
Author(s):  
BO LI ◽  
SANLING YUAN ◽  
WEIGUO ZHANG
Keyword(s):  
Qualitative Analysis ◽  
Incidence Rate ◽  
Computer Simulations ◽  
Epidemic Model ◽  
Mass Action ◽  
Global Dynamics ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Set Up ◽  
Disease Free

In this paper, we study the global dynamics of an epidemic model with nonlinear incidence rate of saturated mass action which depends on the ratio of the number of infectives to that of the susceptibles. The model has set up a challenging issue regarding its dynamics at the R-axis since it is not well defined on it. By carrying out a global qualitative analysis of the model and studying the stabilities of the disease-free equilibrium and the endemic equilibrium, it is shown that either the number of infective individuals tends to zero as time evolves or the disease persists. Computer simulations are presented to illustrate the results.

scholarly journals Analysis of a Deterministic and a Stochastic SIS Epidemic Model with Double Epidemic Hypothesis and Specific Functional Response

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Mouhcine Naim ◽  
Fouad Lahmidi
Keyword(s):  
Epidemic Model ◽  
Deterministic Model ◽  
Sufficient Condition ◽  
Nonlinear Incidence Rate ◽  
Local Asymptotic Stability ◽  
Sis Epidemic Model ◽  
The Stability ◽  
Stochastic Sis Epidemic Model ◽  
Disease Free ◽  
Sis Epidemic

The purpose of this paper is to investigate the stability of a deterministic and stochastic SIS epidemic model with double epidemic hypothesis and specific nonlinear incidence rate. We prove the local asymptotic stability of the equilibria of the deterministic model. Moreover, by constructing a suitable Lyapunov function, we obtain a sufficient condition for the global stability of the disease-free equilibrium. For the stochastic model, we establish global existence and positivity of the solution. Thereafter, stochastic stability of the disease-free equilibrium in almost sure exponential and pth moment exponential is investigated. Finally, numerical examples are presented.

scholarly journals Stochastic Dynamics of an SIRS Epidemic Model with Ratio-Dependent Incidence Rate

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yongli Cai ◽  
Xixi Wang ◽  
Weiming Wang ◽  
Min Zhao
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Stochastic Dynamics ◽  
Complex Dynamics ◽  
Deterministic Model ◽  
Mass Action ◽  
Sirs Epidemic Model ◽  
Stochastic Epidemic ◽  
The Stability ◽  
Disease Free

We investigate the complex dynamics of an epidemic model with nonlinear incidence rate of saturated mass action which depends on the ratio of the number of infectious individuals to that of susceptible individuals. We first deal with the boundedness, dissipation, persistence, and the stability of the disease-free and endemic points of the deterministic model. And then we prove the existence and uniqueness of the global positive solutions, stochastic boundedness, and permanence for the stochastic epidemic model. Furthermore, we perform some numerical examples to validate the analytical findings. Needless to say, both deterministic and stochastic epidemic models have their important roles.

Backward Bifurcation in a Fractional-Order SIRS Epidemic Model with a Nonlinear Incidence Rate

2016 ◽  
Vol 17 (7-8) ◽  
pp. 401-412 ◽  
Author(s):  
A. M. Yousef ◽  
S. M. Salman
Keyword(s):  
Incidence Rate ◽  
Fractional Order ◽  
Epidemic Model ◽  
Local Stability ◽  
Host Population ◽  
Backward Bifurcation ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Sirs Epidemic Model ◽  
Disease Free

Abstract:In this work we study a fractional-order susceptible-infective-recovered-susceptible (SIRS) epidemic model with a nonlinear incidence rate. The incidence is assumed to be a convex function with respect to the infective class of a host population. Local and uniform stability analysis of the disease-free equilibrium is investigated. The conditions for the existence of endemic equilibria (EE) are given. Local stability of the EE is discussed. Conditions for the existence of Hopf bifurcation at the EE are given. Most importantly, conditions ensuring that the system exhibits backward bifurcation are provided. Numerical simulations are performed to verify the correctness of results obtained analytically.

A SIS Epidemic Model with Eventual Impulsive Effects

2013 ◽  
Vol 393 ◽  
pp. 666-674
Author(s):  
Manuel de La Sen ◽  
A. Ibeas ◽  
S. Alonso-Quesada
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Total Population ◽  
Disease Model ◽  
Impulsive Effects ◽  
Time Varying ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Sis Epidemic Model ◽  
Sis Epidemic

This paper studies a time-varyingSIS(i.e.containing susceptible and infected populations) propagation disease model exhibiting a nonlinear incidence rate and impulsive eventual culling of both populations so that the individuals recover with no immunity to the disease. The nonlinear incidence rate consists of two time-varying additive terms proportional to the susceptible and infected populations normalized to the total population.

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